Remark 9.1 Solutions for the parameter \(a\)  from Conjecture REF  have neither zeroes nor poles on
the positive semiaxis. To prove the absence of poles one can surely apply Zhou's vanishing lemma [1]}.
In our case the Riemann-Hilbert problem corresponding to the monodromy data defined in Section 
can be formulated on the positive real semiaxis and the circle centered at 0. The jump matrices are:
\(\begin{pmatrix}1&s_0\\s_0&1+s_0^2\end{pmatrix},\qquad e^{2\pi (a-i/2)\sigma _3},\quad {\rm and}\quad G\) 
